Stone表现定理与广义空间理论

Stone's Representation Theorem and Generalized Space Theory

Ｓｔｏｎｅ表现定理揭示了格论与拓扑空间理论之间的深刻联系，该定理表明可以从纯代数的角度出发而得到若干不同类型的拓扑空间，并可用格论的方法与技巧对拓扑空间的特性进行研究，从而得出关于拓扑学中带有普遍意义的结论。广义空间理论就是沿此方面而诞生的新学科。拓扑分子格理论则是与此相关的又一新学科。

This paper is a survey of the theory of generalized spaces where by generalized spaces we mean Frame theory as well as the theory of topological molecular lattices. It is worthy to note that this paper first offers a general convergence theory in topological molecular lattices.Definition. Let (L(M),μ) be a TML, B ∈ L,S= {A(n),n∈D} be a net in L. We say that S converges to B, in symols S→ B or B= lims, if (1) x ∈M, and x ≤B-, n∈D, there exists A(n,x) such that A(n,x) ≤A(n) and Sx= {A(n,x),n∈ D} converges to x, i. e., every molecular net consisting of molecules selected from A(n) for each n ∈D converges to x, and the convergences are uniform with respect .to x in B-.(2) y ∈ M and y B-,any molecular net mentioned in (1 ) does not converge to y.(3) n ∈ D,A(n) = {A(n,x):x∈M and x ≤ B-}.